To carry on reading about the vacuum, go to ‘Into the void’ on page 110.Our lives have been transformed by knowledge gleaned from scientific discoveries and the technologies they have prompted. It would be nice to think that these advances emerged from a straightforward path between questions being asked and solutions presenting themselves. But that is rarely the case. Scientists may not even know which questions to ask of a new phenomenon, and technologies are sometimes used for a long time before we realise how they work. Yet successes there are, often with their own built-in surprises. Here are a few.
The philosopher Bertrand Russell wrote that ‘almost every serious intellectual advance has had to begin with an attack on some Aristotelian doctrine’. A good example of this is Aristotle’s insistence that the vacuum could not exist. Only in the 17th century, when people had seriously started to ask questions about the world around them, did a simple experiment overturn this thinking by revealing unequivocally that the void is a real, reproducible thing. Physicist Per Eklund traces the story to the present day.
Among the torrent of scientific ideas and technological advances of the 17th century, four are often singled out as ‘great inventions’ for their profound impact on the way humanity views reality. The telescope helped us to understand the solar system and that Earth is not at its centre. The microscope opened up a rich yet previously invisible ‘microworld’ and paved the way for modern medicine, materials technology and much more. The pendulum clock was our first accurate timekeeper and a fundament of modern society.
Perhaps the least well known of the four is the vacuum pump. Its impact is less obvious than the others, possibly because changes in pressure are more difficult to appreciate than images seen through an eyepiece and are not as obviously useful as shared time. What’s more, uses for the vacuum were not immediately forthcoming. Only in the past 150 years has the vacuum become critical in shaping our view of matter and the way we live.
Aristotle’s idea that ‘nature abhors a vacuum’ is one we might smile at today in a world where vacuums exist in everything from food packaging and vacuum cleaners to the Large Hadron Collider. Yet in 1600, almost 2,000 years after Aristotle lived, his view held firm that a vacuum, or empty space, could not exist. Nature, he had argued, was comprised of four elements (earth, wind, fire and water), and space could not be defined where there was nothing. That view began to change following groundbreaking experiments.
The crucial experiments that challenged the status quo were carried out by the Italians Gasparo Berti in 1640 and Evangelista Torricelli in 1644. In Torricelli’s experiment, he filled a metre-long test tube with mercury and turned it upside down in a basin of mercury. The height of mercury in the tube immediately fell to 760 millimetres leaving a gap at the top that could only be a vacuum.
Berti conducted his experiment first, using water instead of mercury, but he was less successful in convincing contemporary natural philosophers of his case. Torricelli is therefore usually given the historic honour of ‘discovering’ the vacuum. In doing so he also unwittingly invented the mercury barometer. A few years later, the Frenchman Blaise Pascal made the first pressure measurement using Torricelli’s set-up – the mercury column rose or fell depending on the atmospheric pressure. And at about the same time, Otto van Guericke in Germany constructed an ‘air pump’, a manually controlled wooden piston pump, rather like a bicycle pump in reverse, that we know today as the first vacuum pump.
One might think that now vacuums were philosophically feasible and technologically possible, there would be a race to reach ever-lower pressures. But this did not happen. Progress halted for some 200 years. Apart from a few inquisitive natural philosophers, nobody could think what to do with the vacuum, so it became something of a sideshow. By 1850, the base pressure that could be reached had been reduced to the vicinity of 1 or 2 millibar, compared with the 6 mbar attained by Robert Boyle in 1660. (To get some feeling for these pressures, see box, ‘What is low pressure like?’)
It is much more difficult to comprehend a low pressure than, say, a small distance. Vacuums encountered in our daily lives are still close to atmospheric pressure. For example, 1 bar is the atmospheric pressure at sea level; a household vacuum cleaner achieves around 750 millibar; and the pressure outside a plane at 10,000 metres altitude is about 250 mbar. But even very simple vacuum pumps reach pressures of the order of 1 mbar.
Picture a milk carton with a volume of 1 litre – that’s one-thousandth of a cubic metres – filled with air at atmospheric pressure. A pressure of 1 mbar corresponds to the same amount of air in 1 cubic metre. If the cube is 10 metres on a side, the pressure is 10–3 mbar, and 10–6 mbar would correspond to a 100-metre-sided cube, larger than any skyscraper.
Two of the most important units of measurement for pressure are named after the earliest contributors to vacuum technology. Evangelista Torricelli’s column of mercury gave rise to the logical pressure unit: 1 millimetre of mercury (mm Hg), or a ‘Torr’. The SI unit for pressure is the Pascal (Pa), after Blaise Pascal who first measured pressure using Torricelli’s set-up. In vacuum science and technology both mbar and Torr are more extensively used than the SI unit. 1 mbar = 0.75 Torr = 100 Pa. Normal air pressure is 1,013 mbar or 760 Torr.
The first major pressure drop came in the mid-1800s, with a mercury pump still based on Torricelli’s principle. This pump, designed by the German Heinrich Geissler in 1855 and then improved on by his compatriot Herman Sprengel, was a hand-driven mercury displacement pump. It forced mercury to drop from a capillary tube at the top of a flask into another at the bottom. Consecutive drops trapped between them tiny amounts of air from the flask, driving down its pressure. Sprengel’s apparently simple set-up improved the attainable pressure by a breathtaking six orders of magnitude to reach 10–5 mbar.
This improvement is an early example of research driven by need. The pumps grew out of the requirements of industry and scientists such as Thomas Edison in the second half of the 19th century. Edison needed a way to stop the filaments in his incandescent light bulbs from burning up. Creating a vacuum inside the bulb solved the problem, and the Sprengel pump found rapid use in his factory.
Likewise, studies of gas discharges in glass tubes, and what we today call plasmas, would not have been possible without Sprengel’s pump. These led to exploration of ‘cathode rays’, which culminated in the discovery of the electron in 1898. Wilhelm Röntgen used vacuum tubes to create the first X-rays, as did Heinrich Hertz in his discovery of the photoelectric effect. These experiments also dispelled the foggy picture of the nature of a vacuum, enhancing the notion that all matter – including rarefied gases at low pressure – is made up of atoms and molecules.
Largely thanks to these advances, the foundations of modern vacuum physics were laid down in the early 1900s. The first of two key breakthroughs was Danish physicist Martin Knudsen’s categorisation of the mechanisms of gas flow at reduced pressure through long narrow tubes. Vacuums today are defined according to his three regimes: a ‘molecular’ flow regime, where the gas is so diluted that molecules interact with each other individually rather than collectively; an intermediate regime; and a ‘viscous’ regime at pressures within a few orders of magnitude of atmospheric, where the gas behaves like a regular fluid. Each regime is governed by fundamentally different physics, and this insight was critical in developing modern vacuum science.
The second breakthrough was Wolfgang Gaede’s pump, created in 1905. It employed a mechanism similar to Sprengel’s in trapping air between drops of mercury, but used a rotating mechanism, which allowed it to be mechanically driven. Gaede’s pump was an order of magnitude faster than Sprengel’s and was the first to require a ‘backing pump’, which kept the outlet below atmospheric pressure. He had invented a pump whose ultimate achievable pressure was so low that the pump could not be connected directly to atmospheric pressure. Backing pumps are now common.
Gaede continued to work on novel pump designs, and in 1915 he presented the first diffusion pump, a type that is still a reliable industrial workhorse today. Despite its title, the pump works not by diffusion but by transferring momentum. Gaede directed a high-speed jet of mercury vapour to propel gas molecules towards the pump’s exhaust, like a game of atomic billiards. Modern-day diffusion pumps use various oils instead of mercury, but the design is essentially the same. The oil diffusion pump came a decade after Gaede’s invention and could reach 10–7 or 10–8 mbar of base pressure.
In the second half of the 20th century, two main needs drove the development of vacuum technology. Thriving high-tech industry, especially making semiconductors for electronics, needed reliable, fast, clean vacuum pumps and chambers for large-scale processing. Yet the main driver for the lowest attainable pressure was the growth of ‘big science’. The competition for nuclear weapons and space exploration that characterised the Cold War, and the particle accelerators used in peaceful research into nuclear and particle physics, both demanded high vacuums.
A key invention from this period is the 1957 turbo-molecular pump, or ‘turbo’ as it is affectionately known to vacuum scientists. Similar in appearance to a jet engine, it ‘bats’ gas molecules towards the pump’s exhaust with rapidly rotating rotor blades. The velocity of the blades is critical: to generate the required momentum, a typical turbo runs at 30,000 to 75,000 revolutions a minute. A turbo can reach pressures as low as about 10–10 mbar, but it cannot go lower because the lighter gases, especially hydrogen, never gain the necessary momentum to be expelled.
Will we ever be able to reach a perfect vacuum? The answer is simple: no. Even if a ‘macroscopic’ volume contained no particles, it would still not be truly ‘empty’ because it would remain subject to quantum fluctuations, dark energy, and other quantum-mechanical phenomena. I think the question should rather be rephrased as ‘will we ever be able to reach a vacuum that, for all practical intents and purposes, would constitute an ideal vacuum?’ This can be done, and most likely has already been done.
In a technological vacuum on Earth, the practical limits are much higher than in outer space. No wall or seal is perfect, and permeation of molecules, especially hydrogen, will always result in there being some particles present. This is especially problematic because most ultra-high-vacuum systems use turbos, which are inefficient at pumping hydrogen. For the most advanced applications in surface science such as scanning tunnelling microscopy and large-scale facilities such as synchrotrons and particle accelerators, this will remain insufficient.
To improve, the base pressure can be pushed down by cryopumping and gettering. Instead of pumping the gas out of the vacuum system, gettering and cryopumping capture the gas molecules inside the system – sticking them to the vacuum chamber itself. A getter is a highly reactive material, such as titanium, that attracts and adsorbs, or binds, any gas left in a vacuum. Coating exposed areas inside a vacuum chamber with a getter can reduce the pressure generated by a turbo pump by as much as an extra two orders of magnitude.
Only cryopumping, however, has the potential to reach an ‘ultimately low’ pressure. As the name suggests, cryopumping works by cooling exposed areas so they adsorb or ‘freeze out’ any remaining gas molecules. Most cryopumps are cooled by liquid nitrogen and are used in industrial applications to reach similar pressures to those achieved by diffusion pumps. But cryopumping can in principle be used to reach much lower pressures. Just how much lower was suggested by Canadian physicist Peter Hobson’s thought experiment on cryopumping about 40 years ago. He assumed an ideal situation in which the vacuum container would be a sealed half-litre glass flask, which would then be immersed in liquid helium at a temperature just 4 degrees above absolute zero. By extrapolating the adsorption curves of gases on glass to the liquid helium temperature, he concluded that ‘it is quite practical to create a pressure of 10–33 [mbar].’ That is, a pressure of some 20 orders of magnitude lower than in any existing vacuum system at the time.
There is a big conceptual problem with this set-up, however. One gas particle in a half-litre corresponds to a pressure of about 10–20 mbar, so how could we possibly end up at 10–33 mbar? This is a neat illustration of the dangers of extrapolation, since achieving that pressure in half a litre would require splitting one molecule into 1013 parts!
Nevertheless, the idea of freezing out gases works and has been used at large-scale facilities such as CERN to store antimatter. When antimatter comes into contact with ordinary matter, they annihilate each other with an ‘explosion’, so antiparticles must be kept away from ordinary particles. This can be achieved by guiding antiparticles, such as antiprotons, into a vacuum container that is then enclosed and immersed in liquid helium.
Antimatter can be stored for months at a time under such conditions. Although there is no way to measure the actual pressure in the container, one can calculate that the particle density must be lower than about 100 atoms per cubic centimetre, which corresponds to 10–16 mbar, to avoid contact between particles and antiparticles. To my mind, this is the ultimate test of a vacuum. The pressure in this vacuum is so low that practically speaking it is a close approximation of a ‘perfect’ vacuum.
Reaching such pressures needs costly and advanced equipment. Commercially, reasonable results can be achieved with cheaper and less complex set-ups. In most practical situations, obtaining a better vacuum is not a goal in itself. It only needs to be as good as is required for the purpose – productivity and cost are more important. Throughout the history of the vacuum, such commercial and technological considerations have been dominant, and there’s little doubt they will remain so into the future.
To carry on reading about the vacuum, go to ‘Into the void’ on page 110.
By any measure some animals are lazy: hanging or lying around all day doing diddly-squat. Sloths are well known for it, and snakes come a close second. But why do they do nothing? When scientists investigated this question, it became clear that these critters have no option. Jonathan Knight finds out why.
After a hard day’s work, the journey home, cooking, washing up and putting the kids to bed, it’s time to collapse into an armchair for five minutes. And in those few snatched moments, you might well envy the unbelievably easy life of some vertebrates. Take the sloth: it sits motionless for hours up a tree in the rainforest canopy. Or the giant python, which lies around for months waiting for its next meal and then rests in the bushes for weeks doing nothing but digesting. Bliss.
Well, not quite. Research into the behaviour and metabolism of such seemingly shiftless animals shows that doing nothing has nothing to do with taking it easy. These animals are operating on the very edge of survival where doing nothing is essential for staying alive. Even more surprisingly, the metabolism of some of the most immobile creatures may be working as hard as a racehorse on the big day.
Mark Chappell, a biologist at the University of California, Riverside, has a particular interest in the energy consumption of animals that live in extreme environments. While working in the Antarctic a couple of decades ago, he discovered that Adélie penguin chicks are not as idle as they seem. Other than occasional brief bouts of begging food from their parents, these juveniles stay fixed to the same spot on the ice for weeks. But when Chappell measured the metabolism of these little birds, he got a shock.
He placed chicks in small sealed chambers with a monitored air supply to find out how fast they were using up oxygen. Oxygen consumption correlates directly with metabolic rate: the higher the demands made on cells, the more oxygen they need to burn glucose and produce energy. Chicks with empty stomachs have a metabolic rate of 1 millilitre of oxygen per gram of body weight per hour.
What really surprised Chappell was that the metabolic rate of freshly fed chicks was twice that figure. Such an increase in metabolic rate is unusually high among warmblooded animals.
When resting, humans have a metabolic rate of about 0.3 millilitres of oxygen per gram of body weight per hour. Light exercise, such as walking, can double this, whereas sprinting can raise it as much as tenfold over short periods. But, at best, digestion drives our metabolism to only about 50 per cent above its resting rate. The rates for most mammals are similar. So a penguin chick digesting its dinner is actually working its metabolism as hard as a human on a brisk walk.
Exercise consumes energy mostly through the working of muscles, but the costs of digestion are more diverse. With the penguin chick’s high-protein diet, about half the energy goes on moving the food along the gut, producing digestive enzymes to break down the food, and pumping the resulting molecules into cells in the gut wall. The other half is used within these cells to reassemble amino acids from the food back into proteins.
But why do penguin chicks expend so much energy on these activities? At this early stage of life, penguins have one objective: to grow as quickly as possible. Chicks make good snacks for skuas, predatory gulls that harass the penguins. In this environment, a small, weak chick is a dead chick. So building up body mass fast is vital for survival, and rapid digestion helps this along. If the chicks bolt their food, they can go back to their parents more frequently for more.
So a souped-up metabolic link to the digestive system helps the chicks build up their bodies quickly. This is why idle ways are so useful: energy wasted on movement cannot be converted into body mass. And, as any couch potato knows, doing nothing is a great way to grow fast.
Chappell found still more surprises about the balance between metabolism, exercise and digestion when he studied another bird that lives much closer to home. House wrens nest in tree holes throughout North America. Their chicks, which hatch blind and helpless, sit still for two weeks, converting whatever the parents feed them into bone, muscle, fat and feathers. In that time, they increase their mass nearly tenfold.
To look at their metabolic capabilities, Chappell put baby house wrens in a sealed chamber with a supply of oxygen and a meter to show how much oxygen they used. He tried a number of different conditions, studying well-fed chicks, hungry chicks and hungry chicks that were prodded to keep them moving around in the nest. His results show that during the first eight days nothing the birds did was more strenuous than digesting food.
A six-day old nestling, for example, had a resting rate of 1 millilitre of oxygen per gram of body weight per hour. In its most energetic burst of activity, it could raise that rate by only half again. But by simply sitting still and digesting, a chick could double this rate and then some – an increase even larger than that of the penguin chicks.
Like humans and other mammals, the chicks’ parents can double their metabolic rate only when they physically exert themselves. By contrast, the chicks are adapted to draw on this extra energy only when digesting food. They are lazy little growth machines designed to work as hard at digesting food as their parents are at bringing it. Yet after about eight days of doing nothing, Chappell found that the house wren chicks’ metabolism flipped so they could consume energy faster during physical activity than during digestion.
In an even more remarkable example of working hard at doing nothing, the Burmese python stays completely still for weeks at a time. Yet the metabolic abilities of this critter rival those of a racehorse going flat out. In his labs at the University of California, Los Angeles, and later at the University of Alabama in Tuscaloosa, Stephen Secor has spent years measuring the rate of oxygen consumption of young Burmese pythons while they are digesting a meal or fasting. The more they eat, he finds, the faster their metabolism.
At a push a small python can eat five rats at a sitting – equal to the snake’s own body mass. Snakes that eat this much can increase their metabolism a stunning 44-fold in less than a day. ‘It looks like they are just sitting there, but they are really huffing and puffing,’ he says. As the meal is digested, which for big meals can take a couple of weeks, the snakes gradually throttle back again. The only other animal ever measured with a metabolic rate running so far above its resting rate is a thoroughbred horse at full gallop.
Of course, in absolute terms the snake has a much lower metabolic rate than the horse – about an eleventh of its rate, in fact. The snake starts with the very low resting rate of about 0.032 millilitres of oxygen per gram of body weight per hour. Like all cold-blooded animals, a python does not have to maintain a constant body temperature the way mammals and birds do, so it expends much less energy at rest. And a python also feeds so infrequently that to conserve energy it shuts down its gut in between meals.
Digestive tracts are very expensive to maintain, mainly because the cells in contact with the food and digestive juices constantly die and slough off. Replacing them takes energy. But the python stops the digestive juices flowing and temporarily interrupts the cells’ replacement cycle. Its gut actually deflates along its length. ‘This is like turning off a car in a traffic jam to save gas,’ says Secor.
Secor found that other python organs tighten their belts in lean times, too. The liver, kidney and heart all shrink gradually as the belly empties. But within a few days of feeding they can grow by up to 50 per cent. The gall bladder is the only organ found to lose weight after a meal as it empties stored-up bile into the gut.
These metabolic adaptations are specifically suited to predators that wait to ambush large, rare prey. For the python, which may go for months without finding a meal, keeping still is a matter of life and death. Moving about would not yield enough extra food to justify the expense, because prey is scarce, and would run away if chased, or both. So a python that constantly went hunting would probably die of starvation.
The Burmese python is not alone in adapting to a niche in which the costs of motion outweigh the benefits, says Brian McNab, emeritus professor of biology at the University of Florida and author of The Physiological Ecology of Vertebrates: A View from Energetics, which explores the role of energy consumption in evolution. The Texas blind salamander lives in lightless caves where the only food is detritus that washes through in streams or leaks through the roof. The only animal that could survive here is one with a very low energy consumption.
So the salamander is by necessity a sedentary creature. But it doesn’t just do nothing to survive – it also sees nothing. McNab argues that the creature is descended from lizards that could see, but that the evolutionary pressure to save energy was so strong that it deprived the salamander of its sight.
It is possible that the animal’s ability to see evolved away through lack of pressure to maintain it: chance mutations in genes controlling vision could have stopped them working and wouldn’t have harmed the salamander’s prospects because it didn’t need to be able to see. But McNab argues there was probably a selective pressure to drive it away. It takes a great deal of energy to maintain vision, he says, because there is a rapid turnover of cells in the retina and cornea. So keeping the eyes when they were of no use would have been terribly wasteful. ‘What they are really doing,’ he says, ‘is saving energy.’
Compared with a dark cave, you might think that the lush rainforests of the Amazon contain an inexhaustible supply of tasty comestibles. And yet among the branches live some remarkably low-energy animals. One of them is the three-toed sloth, a creature that sits motionless for so long that algae grow in its fur.
Even at its peak of activity, climbing a tree in what looks like slow motion, the sloth’s metabolic rate never rises above 0.48 millilitres of oxygen per gram of body weight per hour – three times its resting rate. It simply can’t spare any more energy. Although the sloth can eat all the leaves it wants, the energy inside the leaf is very hard to get at. It is mostly bound up in cellulose, which takes more energy to break down than the simpler carbohydrates in fruits and the proteins in insects.
What’s worse, as a defence against the vast array of insects in the canopy, rainforest leaves have evolved defences to make them difficult to digest. They contain high concentrations of tannins, which bind to the leaves’ proteins and prevent them from being broken down in the animal’s gut. This means that much of the energy in the leaf is unavailable. Most leaves also contain powerful toxins such as alkaloids that take energy to inactivate.
In the rainforest, low- and high-energy animals live side by side. Howler monkeys can often be seen hanging listlessly from trees, while capuchins swing playfully from branch to branch stuffing themselves with fruit. Kenneth Glander of Duke University in Durham, North Carolina, has studied New World primates for more than four decades. He says the smaller, lighter capuchins easily outmanoeuvre the howlers, which weigh nearly twice as much at between 4 and 7 kilograms.
In a sense, the capuchins are creating an extreme environment for the howlers by grabbing all the best food. So when capuchins are around, howlers make do with leaves and slow their metabolism down just as the sloths do. Sometimes, they hang still for hours. Because they must spend so much more energy on processing the leaves in their gut, howlers must do nothing just to survive. ‘In most cases,’ says Glander, ‘howlers are operating on a minimal margin of error.’ If they were more active, they would probably die of starvation.
So the next time you sit down to put your feet up, remember this: while the chance to do nothing may seem like an evolutionary bonus that humans missed out on, all the research into vertebrate loafers shows that being built to do nothing is about survival – and comes with a penalty clause. What’s good for the Texas blind salamander could be bad news for you.
To read more about things that do nothing, go to ‘Putting the idle to work’ on page 183.
It’s a curious thought that just about anything that contains a silicon chip owes its existence to the movement of, well, nothing. Richard Webb tells us why.
The sound of New Year’s Eve celebrations drifting up from the Palace Theater did not distract William Shockley. Nor did the few scattered revellers straying through Chicago’s snow-covered streets below. Rarely a mingler, Shockley had more important things on his mind. Barricaded in his room in the art-deco opulence of the Bismarck Hotel, he was thinking, and writing.
Eight days earlier, on 23 December 1947, John Bardeen and Walter Brattain, two of Shockley’s colleagues at Bell Laboratories in Murray Hill, New Jersey, had unveiled a device that would change the world: the first transistor. Today, shrunk to just nanometres across and carved into beds of silicon, these electrical on-off switches mass in their billions on every single computer chip. Without them, there would be no processing of the words, sounds and images that guide our electronic lives. There would be no smartphone, router, printer, home computer, server or internet. There would be no information age.
Bardeen and Brattain’s device, a rather agricultural construction of semiconductor, gold-enwrapped polystyrene and a spaghetti twist of connecting wires, did not look revolutionary, and it would have taken a seer to foretell the full changes it would bring. Even so, those present that December at Bell Labs knew they had uncovered something big. In Shockley’s words, the transistor was a ‘magnificent Christmas present’. Magnificent, but for one thing: no one knew quite how it worked.
Holed up in his Chicago hotel, Shockley needed to change that. As head of Bell Labs’ solid-state physics group, he had been the intellectual driving force behind the transistor, yet Bardeen and Brattain had made the crucial breakthrough largely without him. To reclaim the idea as his own, he needed to go one better.
That meant getting to grips with a curious entity that seemed to control the transistor’s inner workings. Its existence had been recognised two decades earlier, but its true nature had eluded everyone. For good reason: it was not there.
Transistors – both Bardeen and Brattain’s original and those that hum away in computer processors today – depend on the qualities of that odd half-breed of material known as a semiconductor. Sitting on the cusp of electrical conduction and insulation, semiconductors sometimes let currents pass and sometimes resolutely block their passage.
By the early 20th century, some aspects of this dual personality were well documented. For example, the semiconductor galena, or lead sulphide, was known under certain circumstances to form a junction with a metal through which current travelled in only one direction. That had made it briefly popular in early wireless receivers, where a filigree metal probe – a ‘cat’s whisker’ – was tickled across a crystal of galena to find the contact that would transform a rapidly oscillating radio signal into a pulsing direct current that could power a speaker.
This process had to be repeated afresh each time a radio receiver was switched on, which made tuning a time-consuming and sometimes infuriating business. This was symptomatic of all semiconductors’ failings. There seemed little rhyme or reason in their properties; a slight change in temperature or their material make-up could tip them from conduction to insulation and back again. It was tempting to think their caprices might be tamed to make reliable, reproducible electrical switches, but no one could see how.
And so in the radio receivers and telephone and telegraph systems of the 1920s and 1930s – such as those operated by Bell Labs’ parent company, AT&T – vacuum tubes came to reign supreme. They worked by heating an electrode in a vacuum and applying electric fields of varying strength to the stream of electrons emitted, thus controlling the size of the current reaching a second electrode at the far side. Bulky, failure-prone and power-hungry though they were, vacuum tubes were used as switches and amplifying ‘repeaters’ to hoist fading signals out of a sea of static on their long transcontinental journeys.
Even as they did, however, the seeds of their demise and semiconductors’ eventual triumph were being sown. In 1928 Rudolph Peierls, a young Berlin-born Jew, was working as a student of the great pioneer of quantum physics, Werner Heisenberg, in Leipzig. The convolutions of history would later make Peierls one of the UK’s most respected physicists, and pit him against his mentor in the race to develop the first atomic bomb. At the time, though, he was absorbed by a more niggling problem: why were electrical currents in some metals deflected the wrong way when they hit a magnetic field?
To Peierls, the answer was obvious. ‘The point [was] you couldn’t understand solids without using the quantum theory,’ he recalled in a 1977 interview.1 Just as quantum theory dictates that electrons orbiting an atom can’t have just any old energy, but are confined to a series of separate energy states, Peierls showed that within a solid crystal, electrons are shoe-horned into ‘bands’ of allowed energy states. If one of these bands had only a few occupied states, electrons had great freedom to move, and the result was a familiar electron current. But if a band had only a few vacant states, electron movement would be restricted to the occasional hop into a neighbouring empty slot. With most electrons at a standstill, these vacancies would themselves seem to be on the move: mobile ‘absences of electron’ acting for all the world like positive charges – and moving the wrong way in a magnetic field.
Peierls never gave these odd non-entities a name. It was Heisenberg who gave them their slightly off-hand moniker: Löcher – or ‘holes’. And there things rested. The holes were, after all, just a convenient fiction. Electrons were still doing the actual conducting – weren’t they?
Although Peierls’s band calculations were the germ of a consistent, quantum-mechanical way of looking at how electrical conduction happened, no one quite joined up the dots at the time. It was ten years before the rumblings of war would begin to change that.
Radar technology, which involves bouncing radio waves off objects to determine their distance and speed, would become crucial to Allied successes in the latter stages of the Second World War. But radar presented a problem. If the equipment were to fly on bombing missions, it needed to be compact and lightweight. Vacuum tubes no longer cut the mustard. Might the long-neglected semiconductors, for all their failings, be a way forward?
In 1940, a team at Bell Labs led by engineer Russell Ohl was exploring that possibility by attempting to tame the properties of the semiconductor silicon. At the time, silicon’s grouchy and intermittent conduction was thought to be the result of impurities in its crystal structure, so Ohl and his team set about purifying it. One day, a glitch in the purification process produced a silicon rod with a truly bizarre conducting character. One half acted as if dominated by negatively charged carriers: electrons. The other half, though, seemed to contain moving positive charges.
That was odd, but not half as odd as what happened when you lit up the rod. Left to its own devices, the imbalanced silicon did nothing at all. Shine a bright light on it, however, and it flipped into a conducting state, with current flowing from the negative to the positive region.
A little more probing revealed what was going on. Usually, a silicon atom’s four outer electrons are all tied up in bonds to other atoms in the crystal. But on one side of Ohl’s rod, a tiny impurity of phosphorus with its five outer electrons was creating an excess of unattached electrons. On the other, a small amount of boron with just three electrons was causing an electron deficit (see figure overleaf).
Peierls’s holes had suddenly found a role. When kicked into action by the light, electrons were spilling over from the region of their excess to fill the holes in the electron structure introduced by the boron. However passively, it was the presence of an absence of electrons that was causing the silicon rod’s unique behaviour. Ohl named his discovery the positive-negative or ‘p-n’ junction, owing to its two distinct areas of positive and negative charge carriers. Its property of converting light energy into electric current made it, incidentally, the world’s first photovoltaic cell.
Holes on the march
In silicon (Si) crystals, all electrons are bound. But add boron (B) atoms, which have one fewer binding electrons, and ‘holes’ are created. Electrons leap between these holes to generate a current (at top). The ‘holes’ effectively flow in the opposite direction (at bottom)
It was a few years before Shockley got wind of Ohl’s breakthrough. Already a senior member of Bell Labs’ physics team before the war, he had been taken in a very different direction by the hostilities, becoming head of the US navy’s anti-submarine warfare operations research unit. After resurfacing in 1945 to lead Bell’s solid-state physics division, it did not take Shockley long to spot the p-n junction’s potential.
He was fascinated by the thought that, by adding a metal contact above a junction, you might use an external electric field instead of light to control the current across it. In a sufficiently thin layer of n- or p-type silicon, he reasoned, the right sort of voltage would make electrons or holes swarm towards the contact, providing extra carriers of charge that would boost the current flow across the junction. Varying the voltage would vary the current. The result would be an easily controllable, low-power, small-scale amplifier that would smash the vacuum tube out of sight. That was truly a prospect to pique the interest of Shockley’s paymasters.
His first attempts to realise the dream, though, were unsuccessful. ‘Nothing measurable, no measurable results,’ he noted of an early failure. ‘Quite mysterious.’ And with his mind now on the broad sweep of Bell Labs’ solid-state research, Shockley was obliged to leave further investigations to two highly qualified subordinates: Bardeen, a thoughtful theorist, and Brattain, an inveterate tinkerer.
It proved a frustrating chase, and it was a classic combination of experimental nous and luck that led the pair to success – plus Bardeen’s spur-of-the-moment decision to abandon silicon for its slightly more predictable semiconducting sister germanium. This finally produced the right sort of amplification effect, boosting the power of input signals, sometimes by a factor of hundreds. The magnificent Christmas present was unwrapped.
Just one thing didn’t add up: the current was moving through the device in the wrong direction. Although the germanium slab had n-type material at the top, it appeared to be positive charges making the running. The puzzlement is almost palpable in Brattain’s lab-book entry for 8 December 1947: ‘Bardeen suggests that the surface field is so strong that one is actually getting p-type conduction near the surface,’ he wrote. It was a mental block that stopped Bardeen and Brattain understanding the fruits of their labours.
No doubt they would have done, given time. But in his Chicago hotel room that New Year’s Eve, Shockley stole a march on his colleagues. There was a way out of the impasse, he realised, and he did the first hurried calculations to firm up his case.
If a hole were merely the absence of an electron, then electrons and holes could hardly co-exist: whenever an electron met a hole, its presence would by definition negate the absence of itself that was the hole. By that measure, the existence of positive charges in a negative region, as Bardeen and Brattain had seemingly observed, was a nonsense.
But what if a hole were real, Shockley asked: not just an absence of something, but a true nothing-that-is? What if it were a particle all on its own, with an independent existence just as real as the electron’s? If this were true, holes would not need to fear encountering an electron. They could happily co-exist with electrons in areas dominated by them – and that would explain what was going on in the transistor.
It was a daring intellectual leap. In the weeks that followed, Shockley used the idea to develop a transistor that exploited the independence of electrons and holes. This was the ‘p-n-p’ transistor, in which a region of electron excess was sandwiched between two hole-dominated areas. Apply the right voltage, and the resistance of the middle section could be broken down, allowing holes to pass through hostile electron-populated territory without being swallowed up. It also worked in reverse: electrons could be made to flow through a central region given over to holes. This was the principle that came to underpin the workings of commercial transistors in the decades that followed.
The rest, as they say, is history. For Shockley, it was not a happy one. He did not at first tell Bardeen and Brattain of his new course, and even attempted to claim sole patent rights over the first transistor.2 The relationship between the three men never recovered. By the time they shared the Nobel prize in physics for their discovery in 1956, Shockley had left Bell Labs to form the Shockley Semiconductor Laboratory to capitalise on his transistor alone. But his high-handed and increasingly paranoid behaviour soon led to a mass mutiny from the bright young talents he had hired, such as Gordon Moore and Robert Noyce, who went on to found Intel, which remains the world’s largest manufacturer of microchips.
The hole, meanwhile, went from strength to strength. Today you will find it at the heart of not just every computer chip, but every energy-saving LED lightbulb, every laser that reads our CDs and DVDs, and every touchscreen. Modern life has become unimaginable without this curiosity whose nature took two decades to reveal: the nothing that became a something and changed the world.
As they delve deeper into a topic, scientists regularly find that things are more complicated than they first thought. Genetic screening, for example, has revealed that what we call breast cancer is in fact many distinct diseases. The same thing has happened in space research. What we call the ‘vacuum of space’ turns out to be not one thing but many. Nigel Henbest charts the voids beyond our atmosphere.
In space, no one can hear you scream. There’s nothing out there for sound to travel through. Puncture the skin of the International Space Station, and you’d better seal it off pretty quickly or you’ll soon be breathing a lot of nothing, in a pressure so low your blood will boil.
And yet, you only have to look at the Hubble Space Telescope’s spectacular pictures of gas clouds in space to realise that emptiness is relative. Outer space is not a perfect vacuum, nor are all cosmic vacuums equal. In some places space is teeming with atoms, relatively speaking, while neighbouring regions are much emptier. In recent years, astronomers armed with radio telescopes on the ground and a battery of other kinds of telescope in orbit have been mapping the different vacuums in space. The question is, can they find anywhere in the universe where there really is nothing at all?
It’s not far from Earth’s cocooned surface to the beginning of the cosmic vacuum. As British astrophysicist Fred Hoyle once remarked, ‘Space is only a two-hour drive away, if your car could go vertically upwards.’ And during those first two hours, you would pass through more gas than in the trillions of years it would take you to travel the remaining distance to the edge of the universe.
For human beings, the region where the International Space Station orbits at 400 kilometres above sea level is as near a true vacuum as makes no difference. You wouldn’t last long there without an air supply and pressure suit. Yet the vacuum is not particularly high on a cosmic scale. The air here is still dense enough for astronauts to see a cosmic St Elmo’s fire as their spacecraft rips through the tenuous gas at hypersonic speed.
The residual atmosphere up here creates a slight but potentially lethal wind drag on an orbiting spacecraft. One famous casualty was the early US space station Skylab. Launched in 1973, Skylab was slowed by a braking force from this tenuous gas strong enough to make it spiral downwards and burn up only six years later. The International Space Station is continually boosted to save it from a similar fate.
Earth orbit is sometimes touted as a natural laboratory for studying how things behave when both gravity and atmosphere are absent. In fact, there are residues of both. On the International Space Station, thrusters and moving astronauts produce accelerations equivalent to at least several millionths of Earth’s gravity. And the ‘vacuum’ outside is nowhere near as good as can be achieved by the best laboratory pumps back on Earth.
Even so, there’s something symbolic about the ‘vacuum of space’. In February 2011, space-walking astronaut Al Drew filled a metal cylinder with ‘space’ outside the International Space Station, and brought it back to Earth. This cosmic Message in a Bottle was designed by the Japanese Space Agency to inspire children, as ‘a conduit between humans and space, and between this world and the one beyond us’.
OK, let’s be less romantic for a moment and put these different ‘vacuums’ into figures. The range of gas densities across the universe, from Earth’s atmosphere to the most tenuous cosmic gas clouds, encompasses so many orders of magnitude that we’d soon get lost in the trillions and billionths. Instead, think in terms of how far apart the atoms or molecules are: the higher the vacuum, the larger the average separation of the gas particles.
At sea level on Earth, air molecules are jostling so closely that they are typically just a millionth of a millimetre apart – only a few times the size of the molecules themselves. In Earth orbit, nature stretches the separation of the molecules to around an average one hundredth of a millimetre. What’s more, orbiting vehicles are moving so fast that what the surrounding atoms lack in number they make up in relative speed, as they collide with the craft at 28,000 kilometres per hour.
But there is a way of putting this speed to use to create a high vacuum. In an experiment called Wake Shield, scientists used a simple satellite to smash all records for a human-made vacuum. Wake Shield was essentially a disc of stainless steel, almost four metres across, shaped rather like a saucepan lid. The shield flew on its own several kilometres away from the space shuttle, its convex side forward. As Wake Shield tore through the surrounding gas atoms, it pushed them aside so rapidly that they didn’t have time to diffuse around the back of the satellite. The result was a ‘wake’ of gas atoms behind the steel shield, and a high vacuum at its centre.
At this altitude, Wake Shield increased the average distance between gas atoms to one-tenth of a millimetre. In three space shuttle flights in the 1990s, Wake Shield’s automated lab grew several thin films of semiconductor in the highest artificial vacuum ever achieved – opening the way to ultra-pure chips of new semiconductors and films of high-temperature superconductors.
Go beyond where the International Space Station flies, and Earth’s atmosphere eventually peters out. Compared with planetary atmospheres, the gas in outer space is indeed tenuous. But even so, it is far from empty. No sooner do you rise above Earth’s shroud of air than you enter the atmosphere of the sun. The hot gases in the sun’s outer layer – the corona – constantly boil away into space in a solar wind that sweeps out past the planets.
The blustery solar wind is racked by gusts from magnetic eruptions on the sun’s surface. Although they can light up our skies with magnificent displays of auroras, and disrupt electricity supplies down on Earth, we are talking here about storms in a vacuum. The average density of the solar wind is less even than Wake Shield’s vacuum, with atoms here about a centimetre apart from each other.
Somewhere well beyond the orbit of Neptune, the solar wind has thinned out so far that it is matched by the tenuous gas that fills the space between the stars in the Milky Way. Interstellar gas is invisible even to the best optical telescopes, but it has an accomplice that gives it away. Scattered throughout the gas are tiny dust particles that absorb light from anything lying behind. Where the gas and dust are most concentrated, you see dark clouds in silhouette. To the naked eye, the Coal Sack near the Southern Cross is one familiar example of interstellar matter in bulk. More spectacularly, the Hubble Space Telescope has revealed the vast dusky ‘Pillars of Creation’, sculpted by dark dust silhouetted against the luminous gases of the Eagle Nebula.
Even though these pillars look dense – almost solid – they are extremely tenuous. The dust specks that make them dark are only the size of a particle of cigarette smoke. They are spread so diffusely that you would find only one, on average, in a volume the size of St Peter’s Basilica in Rome. It is only because space is so huge that the dust particles amass to become an all-obscuring fog.
About half the gas in our galaxy lies in relatively dense clouds like these. The rest is spread more widely. Although it is invisible to ordinary telescopes, its hydrogen atoms emit telltale radio energy at a wavelength of 21 centimetres. This interstellar gas is a tangle of dense strands and tenuous patches. If you travelled through the interstellar medium from a tenuous region to a neighbouring denser strand, the density contrast would be greater than diving from Earth’s atmosphere into the sea.
Yet even the denser regions contain atoms a centimetre or more apart. In the tenuous patches, the atoms are another ten times further apart. The difference between the two becomes clear when nature unleashes its ultimate stellar cataclysm – the death of a star in a supernova explosion. Supernovae send out a shock wave that speeds through space in a fireball and shows up brilliantly to telescopes tuned to radio waves and X-rays. The shock wave sweeps up the gas it meets like a snowplough, to form a dense shell.
I have a particular affection for this phenomenon, as it provided my introduction to the various vacuums of space. As a radio astronomer in Cambridge, I was checking out the fine details of the fireball left over from the supernova explosion that was seen from Earth in 1572. It was obvious that the fireball isn’t spherical: the expanding shell has been pushed out of shape as it encountered irregularities in the surrounding medium. Where the surrounding gas is thinner, the shock travels further and faster; where the shock hits a denser strand it slows down. Like a vast natural Wake Shield, the expanding shock wave clears out a ‘vacuum within a vacuum’, leaving the gas inside the supernova remnant thousands of times less dense than even the tenuous interstellar gases outside it. And the passage of the fireball has raised the temperature inside the shell to millions of degrees.
The discovery of hot, tenuous gas within a supernova’s fireball was no surprise. But astronomers were puzzled to find signs of similar hot and extremely tenuous gas in parts of our galaxy, the Milky Way, where there was no sign of recent supernova explosions.
Gas this thin is very hard to study: with so little material, evidence becomes more and more difficult to find. The first sign that it exists came from the imprint of its spectral lines on the ultraviolet light from distant stars. Its existence was confirmed by the discovery of faint X-rays given off by the gas. Astronomers have now concluded that half the galaxy’s volume is filled with million-degree gas whose individual atoms – in fact ions, as they are stripped of their electrons – are fully 10 centimetres apart.
This hot gas, it seems, has come from thousands of supernova explosions over the aeons, each blowing its own bubble in the surrounding interstellar gas. And as the bubbles grew, they coalesced to produce a galaxy-sized ‘Swiss cheese’ – a network of holes surrounded by denser matter. The hot gas has also burst out of its colder surroundings, and now envelops the Milky Way in a faintly glowing halo.
It’s a good vacuum – the best in the galaxy – but it’s still not perfect. How about the reaches of space beyond? In general, galaxies are not spread uniformly through space. Most of them are gathered in giant clusters, and here you might expect to find the denser parts of the inter-galactic medium. Indeed, X-ray telescopes have picked out pools of gas at temperatures of 100 million degrees or more, bound by the gravity of the galaxies in a cluster. The density of these pools is similar to that of the gas in the most tenuous parts of our galaxy, with a distance between ions of about 10 centimetres.
Astronomers are only beginning to probe the inter-galactic regions where matter is spread even more thinly. It’s too much to hope that the gas here will emit any significant amount of radiation. But that does not mean it is undetectable. The atoms may reveal themselves by absorbing radiation coming from bright beacons beyond. And nature has provided astronomers with suitable light sources: distant and extremely bright galaxies called quasars.
Intergalactic gas is pulled and pummelled by the gravity of galaxies and the enigmatic ‘dark matter’, which is invisible to conventional telescopes yet is thought to make up about 80 per cent of all matter. As a result, it should be heated to around a million degrees: a state that astronomers call ‘warm-hot intergalactic medium’, or WHIM for short. As radiation from distant quasars passes through, the WHIM should imprint a dark pattern of absorption lines in the X-ray part of the spectrum.
Using results from two orbiting X-ray telescopes – the European XMM-Newton and NASA’s Chandra satellite – astronomers discovered exactly this pattern of lines in three different directions in space. In each case, the WHIM was pooled up in a ‘wall’ of galaxies, just as theory suggests.
The atoms of gas confined in these walls are, on average, around 50 centimetres apart. Even so, they cover such vast tracts of space that the clouds of WHIM may contain as much mass as all the visible galaxies in the universe put together.
The filamentary walls of galaxies and WHIM surround huge voids in the cosmos, where matter is far more tenuous: the voids are typically 50 million light years across. But even here we don’t reach a true vacuum. Astronomers have found a smattering of galaxies within the voids, around one-tenth the average density of galaxies in the universe as a whole.
In our quest for the ultimate vacuum, we have literally come as far as we can. Assuming the interstellar gas in the voids is in rough proportion to the galaxies here, then the most tenuous part of our cosmos still contains some atoms. The spacing between them is around 5 metres.
So here, at the ends of the universe, lies the physicists’ dream vacuum. Absolute emptiness. Take an experimental chamber the size of your living room to one of these voids and open the doors. When all the air has gone, there’ll be nothing at all inside – or possibly just a single atom, if you’re very unlucky.
To read more about vacuums, go to ‘The turbulent life of empty space’ on page 126.
We have heard that when it arrived in Europe, zero was treated with suspicion. We don’t think of the absence of sound as a type of sound, so why should the absence of numbers be a number, argued its detractors. It took centuries for zero to gain acceptance. It is certainly not like other numbers. To work with it requires some tough intellectual contortions, as mathematician Ian Stewart explains.
Nothing is more interesting than nothing, nothing is more puzzling than nothing, and nothing is more important than nothing. For mathematicians, nothing is one of their favourite topics, a veritable Pandora’s box of curiosities and paradoxes. What lies at the heart of mathematics? You guessed it: nothing.
Word games like this are almost irresistible when you talk about nothing, but in the case of maths this is cheating slightly. What lies at the heart of maths is related to nothing, but isn’t quite the same thing. ‘Nothing’ is – well, nothing. A void. Total absence of thingness. Zero, however, is definitely a thing. It is a number. It is, in fact, the number you get when you count your oranges and you haven’t got any. And zero has caused mathematicians more heartache, and given them more joy, than any other number.
Zero, as a symbol, is part of the wonderful invention of ‘place notation’. Early notations for numbers were weird and wonderful, a good example being Roman numerals, in which the number 1,998 comes out as MCMXCVIII – one thousand (M) plus one hundred less than a thousand (CM) plus ten less than a hundred (XC) plus five (V) plus one plus one plus one (III). Try doing arithmetic with that lot. So the symbols were used to record numbers, while calculations were done using the abacus, piling up stones in rows in the sand or moving beads on wires.
At some point, somebody got the bright idea of representing the state of a row of beads by a symbol – not our current 1, 2, 3, 4, 5, 6, 7, 8, 9, but something fairly similar. The symbol 9 would represent nine beads in any row – nine thousands, nine hundreds, nine tens, nine units. The symbol’s shape didn’t tell you which, any more than the number of beads on a wire of the abacus did. The distinction was found in the position of the symbol, which corresponded to the position of the wire. In the notation 1,998, for instance, the first 9 means nine hundred and the second ninety.
The idea of place notation made it rather important to have a symbol for an empty row of beads. Without it, you couldn’t tell the difference between 14, 104, 140 and 1,400. So in the beginning the symbol for zero was intimately associated with the concept of emptiness, rather than being a number in its own right. But by the 7th century, that had started to change. The Indian astronomer Brahmagupta explained that multiplying a number by 0 produced 0 and that subtracting 0 from a number left the number intact. By using 0 in arithmetic on the same footing as the other numbers, he showed that 0 had genuine numberhood.
Pandora’s box was now wide open, and what burst forth was – nothing. And what a glorious, untamed, infuriating nothing it was.
The results obtained by doing arithmetic with zero were often curious, so curious sometimes that they had to be forbidden. Addition had the same effect as subtraction: the number stayed the same. Linguistic purists may object that leaving something unchanged hardly amounts to addition, but mathematicians generally prefer convenience to linguistic purity. Multiplication by zero, as Brahmagupta said, always yielded zero. It was with division that the serious trouble set in.
Dividing 0 by a non-zero number is easy: the result is 0. Why? Because 0 divided by 7, say, should be ‘whatever number gives 0 when multiplied by 7’, and 0 is the only thing that fits the bill. But what is 1 divided by 0? It must be ‘whatever number gives 1 when multiplied by 0’. Unfortunately, any number multiplied by 0 gives 0 not 1, so there’s no such number. Division by zero is therefore forbidden, which is why calculators put up an error message if you try it.
Instead of forbidding fractions like 1 divided by 0, it is possible to release yet another irritant from Pandora’s mathematical box – by defining 1 divided by 0 to be ‘infinity’. Infinity is even weirder than zero; its use should always be accompanied by a government warning: ‘Infinity can seriously damage your calculations.’ Whatever infinity may be, it isn’t a number in the usual sense. So mostly it’s best to avoid things like 1 divided by 0.
Sorry: Pandora’s curse is not so easily evaded. What about 0 divided by 0? Now the problem is not an absence of suitable candidates, but an embarrassment of them. Again, 0 divided by 0 should mean ‘whatever number gives 0 when multiplied by 0’. But since this is true whatever number you use to divide 0 by, unless you’re very careful, you can fall into many logical traps – the simplest such being the ‘proof’ that 1 = 2 because both equal 0 when they are divided by 0. So 0 divided by 0 is also forbidden.
Alas, 0 divided by 0 was too seductive to stay forbidden for long. It is at the heart of calculus, the independent invention of Gottfried Wilhelm von Leibniz and Isaac Newton. Calculus was an extraordinary intellectual revolution, perhaps without historical parallel, because it gave us the idea that nature is at root mathematical.
In what sense is calculus about 0 divided by 0? Well, the underlying feature of calculus is the rate of change of some variable – how rapidly it is changing at a given instant. Here a formula or two seems unavoidable. Suppose some quantity x varies with time t, and write x(t) for its value at time t. This x might be how far your bike has travelled, so x(12 noon) = the Pig and Whistle pub. At that point your bike is probably not moving, unless an unfriendly local is stealing it, so the rate of change of x at 12 noon is zero. However, by some time t a bit after 2 pm, you are pedalling along the leafy byways at position x(t). How fast is your position changing, at that precise instant?
Newton’s answer was to let time increase by a tiny amount – let’s call it d. As t changes to t + d, your bike moves from x(t) to x(t + d) – say from level with a dozing sheep’s left nostril to level with its right nostril. The amount by which your position changes is x(t + d)−x(t), the inter-nostril distance, and since it took you time d to achieve that change, the rate of change is (x(t + d) − x(t))/d; distance travelled divided by time taken to do so.
So far so good, but this expression represents the average rate of change over the time interval from t to t + d, not the rate of change at time t itself. However small d may be, even if it’s 0.00000000001, this approach still doesn’t give you the instantaneous rate of change. Newton’s idea was to find the average rate of change over a time interval of length d, let d become zero, and see what you get.
In practice this leads to entirely sensible answers, but the procedure is mysterious. Enter Bishop Berkeley, best known for his philosophical writings on the problem of existence. Berkeley annoyed all the mathematicians by pointing out – correctly – that Newton’s procedure amounts to dividing 0 by 0. Over a time interval of zero, your bike moves a distance of zero, and you’re dividing one by the other.
Berkeley had an ulterior motive: he was upset by criticisms that religious faith was illogical, and he hit back by pointing out that calculus is illogical too. He did so in 1743 in a pamphlet entitled The Analyst, Or a Discourse Addressed to an Infidel Mathematician Wherein it is examined whether the Object, Principles, and Inferences of the Modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith. It contained the following: ‘First cast out the beam in thine own Eye; and then shalt thou see clearly to cast out the mote out of thy brother’s Eye.’ Clearly the good bishop was a bit peeved; equally clearly, he did his homework on the maths.
Newton tried to justify his calculations by appealing to physical intuition, and also by a rather weaselly explanation of how the method avoids dividing by zero. First you write down your equation using the variable d. The fraction involves dividing by d, but that’s all right because at this stage you’re saying that d is not zero. You then simplify your fraction until the d in the denominator disappears. Only then do you let d equal zero to get your answer.
How d can sometimes be allowed to be zero and sometimes not, Newton never really explained. Leibniz made a more mystical appeal to the ‘spirit of finesse’ as opposed to the ‘spirit of logic’ (which loosely translates as ‘I don’t know what I’m doing, but hey, it works’). Berkeley claimed the method worked because of compensating errors, but missed the key point: why do the errors compensate?
In the end, the whole problem was tidied up by Karl Weierstrass, about 120 years later, who defined the elusive concept of a ‘limit’. Rather than saying that d sometimes can and sometimes cannot be zero, you’re actually calculating the value that the fraction approaches as d gets closer and closer to zero. And it works. So Newton and Leibniz created a new way of thinking about the world, while Berkeley’s criticism, though right, was uncreative. The whole dispute, in fact, turned out to be about nothing.
Zero tolerance
Zero is a born troublemaker. Once mathematicians decided to treat it as a number, all the standard formulas had to be extended to embrace zero – with results that were not always intuitive. The most familiar example is powers. Take the fourth power of 5. This is 54 or 5 × 5 × 5 × 5. So clearly 50 must be five to the power of zero, or no fives multiplied together.
This is obviously not the way to think of it. Instead, what mathematicians do is to decide on some property of powers that they want to remain true. For instance, if you multiply powers together, the exponents add up: 52 × 53 = (5 × 5) × (5 × 5 × 5) = 55.
If you want 50 to be any use, it makes sense to retain this property, so that 50 × 52 must equal 50+2 = 52 = 25. That is, 50 × 25 equals 25. Therefore 50 must equal 1. For this reason, the standard convention is that the zeroth power of any number is 1 – with one exception: 00. The above argument requires 00 × 0 to equal 0, sure – but now you can’t divide out the 0s to conclude that 00 = 1. In fact, just like 0 divided by 0, you have to deem 00 meaningless.
The same kind of approach determines the convention for zero factorial. Factorials, symbolised by an exclamation mark, are normally defined like this: 5! = 5 × 4 × 3 × 2 × 1. Starting with the chosen number, reduce it one step at a time until you hit 1, then multiply the resulting numbers together.
But this doesn’t help with 0!, since you have to stop before you start. The usual interpretation of n! is ‘the number of ways to arrange n things in order’, but this also doesn’t help, since it’s not at all clear how many ways there are to arrange no things in order. The most plausible answer would seem to be ‘none, because there aren’t any things to arrange’, but that approach turns out to be misleading. Mathematicians prefer to preserve a general property of factorials, the pattern
4! = 4 × 3!
3! = 3 × 2!
2! = 2 × 1! and extend it to
1! = 1 × 0! Since 1! = 1, this leads to the conclusion that 0! = 1. And this is the convention that mathematicians employ.
To read more about zero, go to ‘Nothing in common’ on page 158.